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Properties

Label 2-34e2-17.13-c1-0-13
Degree $2$
Conductor $1156$
Sign $0.992 + 0.122i$
Analytic cond. $9.23070$
Root an. cond. $3.03820$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.26 + 1.26i)3-s + (−0.559 − 0.559i)5-s + (3.38 − 3.38i)7-s + 0.208i·9-s + (−3.24 + 3.24i)11-s + 1.79·13-s − 1.41i·15-s + 0.208i·19-s + 8.58·21-s + (4.80 − 4.80i)23-s − 4.37i·25-s + (3.53 − 3.53i)27-s + (6.92 + 6.92i)29-s + (3.53 + 3.53i)31-s − 8.20·33-s + ⋯
L(s)  = 1  + (0.731 + 0.731i)3-s + (−0.250 − 0.250i)5-s + (1.28 − 1.28i)7-s + 0.0695i·9-s + (−0.977 + 0.977i)11-s + 0.496·13-s − 0.365i·15-s + 0.0478i·19-s + 1.87·21-s + (1.00 − 1.00i)23-s − 0.874i·25-s + (0.680 − 0.680i)27-s + (1.28 + 1.28i)29-s + (0.635 + 0.635i)31-s − 1.42·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $0.992 + 0.122i$
Analytic conductor: \(9.23070\)
Root analytic conductor: \(3.03820\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :1/2),\ 0.992 + 0.122i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.281705365\)
\(L(\frac12)\) \(\approx\) \(2.281705365\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 + (-1.26 - 1.26i)T + 3iT^{2} \)
5 \( 1 + (0.559 + 0.559i)T + 5iT^{2} \)
7 \( 1 + (-3.38 + 3.38i)T - 7iT^{2} \)
11 \( 1 + (3.24 - 3.24i)T - 11iT^{2} \)
13 \( 1 - 1.79T + 13T^{2} \)
19 \( 1 - 0.208iT - 19T^{2} \)
23 \( 1 + (-4.80 + 4.80i)T - 23iT^{2} \)
29 \( 1 + (-6.92 - 6.92i)T + 29iT^{2} \)
31 \( 1 + (-3.53 - 3.53i)T + 31iT^{2} \)
37 \( 1 + (3.94 + 3.94i)T + 37iT^{2} \)
41 \( 1 + (-4.24 + 4.24i)T - 41iT^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 - 2.20iT - 53T^{2} \)
59 \( 1 - 9.16iT - 59T^{2} \)
61 \( 1 + (-3.83 + 3.83i)T - 61iT^{2} \)
67 \( 1 + 5.58T + 67T^{2} \)
71 \( 1 + (-4.24 - 4.24i)T + 71iT^{2} \)
73 \( 1 + (-8.30 - 8.30i)T + 73iT^{2} \)
79 \( 1 + (-1.41 + 1.41i)T - 79iT^{2} \)
83 \( 1 + 2.20iT - 83T^{2} \)
89 \( 1 + 9.16T + 89T^{2} \)
97 \( 1 + (8.45 + 8.45i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.977677958000405362465881351538, −8.679966157150249795765047909853, −8.381201979264555401321011261852, −7.43953487950766659816461647624, −6.68920908946616047515277610888, −4.95935609007683814989164986272, −4.63032381836929955416356742823, −3.75058830649119998558504152530, −2.57222349132963907635741016582, −1.07783878324355221882519769283, 1.41363716873928686179859979738, 2.51364568826026558120887510340, 3.19610419787389980455006696047, 4.83013200428089339420927181895, 5.50705984094583720483019575674, 6.51787314292321554370995982416, 7.76034007648046363821485829477, 8.121644951088655598171163350295, 8.620647581605876713378027699683, 9.603800464407584537463489622094

Graph of the $Z$-function along the critical line